MultiStream Detection (MuStD) Halo Finder

• MultiStream Detection (MuStD) is a halo finder that defines virialized regions as closed, convex envelopes around local multistream maxima in dark matter simulations.
• It computes the multistream field from a tessellated Lagrangian grid and applies Gaussian smoothing followed by Hessian eigenvalue analysis to segment halo candidates.
• By avoiding arbitrary density thresholds and linking lengths, MuStD offers a dynamical-geometric framework that highlights the history of gravitational collapse.

MultiStream Detection (MuStD) is a halo-identification framework that detects potential dark matter haloes in cosmological $N$-body simulations by analyzing the multistream field $n_{\rm str}(\mathbf{x})$, a scalar field that counts the number of velocity streams at each Eulerian location. In “Dark matter haloes: a multistream view” (Ramachandra et al., 2017), MuStD is formulated around the idea that the multistream field is unique to collisionless dark matter and that virialized haloes correspond to closed, non-percolating, approximately convex regions surrounding local multistream maxima. Rather than imposing a density threshold, a Friends-of-Friends linking length, or a spherical-overdensity radius, the method smooths $n_{\rm str}$, computes the Hessian of $-n_{\rm str}$, and labels regions satisfying a convexity criterion as halo candidates.

1. Physical basis and definition of the multistream field

MuStD is grounded in the phase-space description of collisionless dark matter. In an $N$-body simulation, dark matter may be treated as a continuous three-dimensional “phase-space sheet” that evolves under gravity from an initially nearly uniform state. As structures collapse, different portions of that sheet fold and overlap in physical, or Eulerian, space. At any fixed spatial location $\mathbf{x}$, the number of physically distinct velocity branches is an integer, denoted $n_{\rm str}(\mathbf{x})$ (Ramachandra et al., 2017).

Operationally, the multistream field is constructed by tessellating the initial Lagrangian grid into tetrahedra, advecting the tetrahedral vertices with the $N$-body integrator, and then depositing the tetrahedra onto a regular Eulerian diagnostic grid of spacing $l_d$. Each grid-cell center, or vertex, counts how many tetrahedra cover that point. Because each tetrahedron carries a single-valued velocity, the count of overlapping tetrahedra equals the number of distinct velocity streams there. In symbolic form,

$$n_{\rm str}(\mathbf{x}) \;=\; \sum_k \mathbf{1}_{\mathcal T_k}(\mathbf{x}),$$

where $n_{\rm str}(\mathbf{x})$0 is unity if $n_{\rm str}(\mathbf{x})$1 lies inside tetrahedron $n_{\rm str}(\mathbf{x})$2, and zero otherwise.

This construction gives MuStD a dynamical rather than purely configurational basis. The field $n_{\rm str}(\mathbf{x})$3 emerges at the nonlinear stage of perturbation growth because dark matter is collisionless, and counting the number of velocity streams supplements information obtained from spatial clustering alone. A central distinction is the clean interpretation of $n_{\rm str}(\mathbf{x})$4 as the single-stream regime associated with voids.

2. Convexity hypothesis and Hessian criterion

The geometric premise of MuStD is that, in the virialized stage of collapse, halo boundaries should appear as closed, non-percolating, approximately convex surfaces in the multistream field. Physically, the assumption encodes the expectation that matter inside a halo has overturned in all three directions and therefore has high $n_{\rm str}(\mathbf{x})$5, whereas outside the collapse is incomplete or uni-directional, as in filaments, walls, and voids (Ramachandra et al., 2017).

The method therefore works with the scalar field

$n_{\rm str}(\mathbf{x})$6

and forms its Hessian,

$n_{\rm str}(\mathbf{x})$7

At each grid point, the symmetric $n_{\rm str}(\mathbf{x})$8 matrix $n_{\rm str}(\mathbf{x})$9 is diagonalized, with ordered eigenvalues

$n_{\rm str}$0

A necessary and sufficient local condition for a closed convex “blob” in the field $n_{\rm str}$1 is

$n_{\rm str}$2

Equivalently, all three principal curvatures of $n_{\rm str}$3 are positive. In the original $n_{\rm str}$4 field, this identifies a local maximum direction in all three dimensions. Because the eigenvalues are ordered, the single scalar condition $n_{\rm str}$5 is sufficient to select these convex multistream peaks.

The significance of this criterion is methodological. MuStD does not define haloes as density excursions or percolating linked sets; it isolates regions through local curvature of a field derived from the folded Lagrangian sheet. This suggests a geometrically constrained notion of halohood in which each accepted region encloses one multistream peak.

3. Computational workflow

The MuStD workflow is presented as a stepwise segmentation procedure. First, the raw multistream field $n_{\rm str}$6 is computed from the tessellated Lagrangian grid evolved to $n_{\rm str}$7, with tetrahedra deposited onto a uniform diagnostic grid. Second, the field is smoothed by convolution with a three-dimensional Gaussian of chosen width $n_{\rm str}$8. Third, second-derivative finite differences of $n_{\rm str}$9 are computed on the diagnostic grid to obtain the Hessian, and the ordered eigenvalues $-n_{\rm str}$0 are evaluated at each cell (Ramachandra et al., 2017).

Candidate regions are then segmented from the mask

$-n_{\rm str}$1

using connected-component labeling, or “flood fill,” to produce disjoint labeled regions $-n_{\rm str}$2. Each connected label is required to contain a sufficiently high local stream-count peak: any label with $-n_{\rm str}$3 is discarded. The stated rationale is that this enforces at least three independent foldings, or shell crossings, inside each halo. Surviving regions are then mapped back to the $-n_{\rm str}$4-body particle set, and regions containing fewer than $-n_{\rm str}$5 particles are rejected. The remaining labeled regions constitute the MuStD halo catalogue.

By construction, each output region is a closed, convex envelope around a single $-n_{\rm str}$6 maximum. In a single-scale analysis of high multistream field resolution and low softening length, halo substructures with local multistream maxima are isolated as individual halo sites. This behavior is particularly relevant in crowded environments, where multiple density peaks may occupy a common large-scale structure.

4. Smoothing, resolution, and scale dependence

The raw multistream field is integer-valued and noisy at the grid scale, so MuStD applies Gaussian smoothing before the Hessian analysis. In the reported experiments, $-n_{\rm str}$7 is computed on diagnostic grids of side $-n_{\rm str}$8 with refinement factors $-n_{\rm str}$9 or $N$0, corresponding to $N$1 or $N$2 for a $N$3 or $N$4 run. Gaussian smoothing scales of one grid spacing, $N$5, two grid spacings, $N$6, and larger values are tested (Ramachandra et al., 2017).

The $N$7-body calculations use cosmological gravitational softening lengths $N$8 for the lower-resolution case and $N$9 for the higher-resolution case. These parameters control how sharply the multistream sheet can fold at small scales, although the description states that they have only a weak effect on the multistream counts themselves.

As presented, MuStD is a single-scale method: one chooses a single Gaussian $\mathbf{x}$0 and performs the Hessian-eigenvalue analysis at that scale. The authors explicitly note that a genuinely multi-scale extension, obtained by scanning in $\mathbf{x}$1, would allow recovery of halo-subhalo hierarchies. They also show qualitatively that increasing $\mathbf{x}$2 tends to wash out small subhaloes and increase the convexity of large haloes. A plausible implication is that the scale parameter plays a dual role, controlling both noise suppression and the level of structural granularity retained in the final catalogue.

5. Comparison with standard halo finders

The comparative evaluation in the paper focuses on AHF, FOF, and MuStD in a $\mathbf{x}$3 Mpc box at two particle resolutions. Halo counts and enclosed mass fractions show that MuStD lies between AHF and FOF over much of the tested range, while mass functions agree closely in the knee region $\mathbf{x}$4 (Ramachandra et al., 2017).

Simulation setup	Halo counts: AHF / MuStD / FOF	Halo mass fraction: AHF / MuStD / FOF
$\mathbf{x}$5 particles	$\mathbf{x}$6 / $\mathbf{x}$7 / $\mathbf{x}$8	$\mathbf{x}$9 / $n_{\rm str}(\mathbf{x})$0 / $n_{\rm str}(\mathbf{x})$1
$n_{\rm str}(\mathbf{x})$2 particles	$n_{\rm str}(\mathbf{x})$3 / $n_{\rm str}(\mathbf{x})$4 / $n_{\rm str}(\mathbf{x})$5	$n_{\rm str}(\mathbf{x})$6 / $n_{\rm str}(\mathbf{x})$7 / $n_{\rm str}(\mathbf{x})$8

At the very highest masses, $n_{\rm str}(\mathbf{x})$9, MuStD slightly underpredicts relative to FOF. The explanation given is that the strict convexity test can split or reject very large haloes with substructure. About $N$0 of mass is jointly flagged as halo by all three methods; the differences are attributed to percolation into the single-stream void, convexity splitting, and the absence of any unbinding step in MuStD.

The spatial correspondence between catalogues also differs systematically. MuStD haloes, by design, never overlap voids with $N$1, whereas FOF sometimes bridges into voids. MuStD boundaries are non-spherical but compact, removing tenuous bridges that FOF can leave, while AHF forcibly makes each halo spherical. In crowded filamentary regions, FOF can merge multiple density peaks into one object; MuStD instead splits them because each separate convex envelope is tied to a single $N$2 maximum.

6. Strengths, limitations, and interpretation

MuStD has several explicitly stated strengths. It uses full Lagrangian dynamical information rather than density alone; it distinguishes voids from non-voids through the criterion $N$3; it identifies haloes by a purely local curvature test rather than by an arbitrary linking length or global density threshold; and it naturally isolates substructure wherever separate multistream peaks occur (Ramachandra et al., 2017).

Its limitations are equally specific. A strict convexity requirement can undercount or fragment very massive, strongly substructured haloes. The smoothing scale $N$4 is a free parameter, and a full multi-scale extension is identified as necessary for a natural treatment of halo-subhalo hierarchy. The framework also performs no explicit unbinding of unvirialized particles, so weakly bound outskirts may be excluded if their local curvature fails the $N$5 test.

These features bear directly on interpretation. MuStD is not a density-based halo finder augmented with an auxiliary diagnostic; it is a dynamical-geometric framework whose core observable is the multistream field extracted from the six-dimensional Lagrangian sheet. This suggests a different ontology of halo identification: one centered on shell crossing, local multistream maxima, and convex segmentation. The principal tension in the method is therefore not between different density thresholds, but between the physical appeal of the convexity condition and its tendency to fragment highly structured massive systems.

7. Position within cosmic-web analysis

Within the broader context of cosmic-web analysis, MuStD treats haloes as the highest concentrations of dark matter embedded in a hierarchy of collisionless structures extending from voids to walls and filaments. The multistream field emerges specifically because dark matter is collisionless, and the method exploits that property to classify highly nonlinear structures using information unavailable to purely density-based descriptions (Ramachandra et al., 2017).

The framework is especially notable for making halo detection depend on the topology and curvature of a field derived from the folded phase-space sheet. In that sense, MuStD links halo finding to the dynamical history of gravitational collapse rather than only to the Eulerian snapshot of mass concentration. This suggests a broader research direction in which halo catalogues, substructure catalogues, and cosmic-web morphology are treated as related manifestations of the same multistream geometry.